Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

By Michael Christ, James E. Colliander, Terrence Tao. 2002

464 citations recorded by the source corpus.

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrodinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity H -¾ , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [- T, T ] and a locally Lipschitz continuous map taking the initial data in H -¾ to a distributional solution u ∈ C 0 ([- T, T ]; H -¾ ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5c6c041831ba16de678744a4fb31706ccecb83cb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Nonlinear wave interactions for the Benjamin-Ono equation. (2004, background): “Moreover, essentially all papers [6, 7, 8, 12] deal with the self interaction of a single high frequency wave which naturally lead to upper bounds on the Sobolev regularity s which is consistent with the smooth dependence on the initial data for large s holding in the cases considered in [6, 7, 8 ,”
Sharp well-posedness results for the BBM equation (2008, result): “This result turns out to be sharp in the sense that below this value of s, the initial-value problem cannot be solved by a Picard iteration, as is shown in the recent work of Christ, Colliander and Tao [14] (see also [19] [12] [22] and further comments in the concluding section).”
Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation (2015, unknown): “It is by now known that local well-posedness holds in H for s ≥ −3/4 (see [4,14]) and that for s < −3/4, there is ill-posedness (see [4,15,18]).”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, background): “has been an area of intense interest, and (1.1) in particular. Conjecture 1.2 The initial value problem (1.6) with p= 4 d is globally well - posed and scattering for any u0 ∈ L2(Rd). Remark: [17] and [18] proved that this conjecture is sharp. [87] proved conjecture 1.2 is true for dimensions d≥ 3 wi”
Almost sure well-posedness of the cubic nonlinear Schr\ (2009, methodology): “a(x,t) = aei(Nx+N 2t∓|a|2t) solves (3) for a ∈ C and N ∈ N. Then, we can follow the argument by Burq-G´erard-Tzvetkov [4] to show the ill-posedness of (3) below L2(T). Also, see Christ-Colliander-Tao [6]. Remark 1.3. In studying the invariance of the Gibbs measure for the 2-d defocusing cubic NLS, B”
The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality (2008, methodology): “Additionally, we will use a simple but beautiful idea of asymptotic embedding of a NLS solution to an approximate gKdV solution, which was previously exploited in [7] and [29].”
Long time behavior of solutions to the mKdV (2014, background): “Below s = 1 4 the solution map fails to be uniformly continuous [1, 26] although weaker forms of well-posedness hold [2].”
Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R) (2008, methodology): “The main reason that the H local solution of KdV in [4] can not be extended to a global solution by using the I-method is that the local solution doesn’t have as many properties as the H local solution for s > −3/4.”
KdV is well-posed in H–1 (2018, background): “ proof of local well-posedness for (KdV) in Hs(R) for s≥ −3/4 and in Hs(R/Z) for s≥ −1/2. Excepting the endpoints, this was proved by Kenig–Ponce– Vega in [37]. For a discussion of the endpoints, see [10] and [12, 13, 14]. These ranges of sare sharp if one requires the data to solution map to be uni”
Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “[6], we extend non-uniform dependence on initial data in Sobolev spaces H for CH to the range 3/2 < s < 2.”